Advocaat guided the club to Premier League safety with a game to spare after taking over from the sacked Gus Poyet on March 17.
The 67-year-old was offered a three-year contract at the Stadium of Light, but the Dutchman has decided not to accept.
???At certain moments in your life you have to make a decision and I am almost 68 years old now. A number of clubs contacted me but the decision was always Sunderland or nobody,??? Advocaat told the club???s website.
???I would like to thank the chairman, Lee, the staff and of course the fans. We gave everything in our time there and achieved what we set out to. Sunderland has been one of the highlights of my career.???
Sunderland chairman Ellis Short said: ???Although I am extremely disappointed, I understand and absolutely respect his decision. In his brief time here he made a real impact and lifted the club with his experience and enthusiasm. We will always be grateful to him for what he achieved.
The club???s sporting director Lee Congerton added: ???I know it was a tough decision for Dick to make but he felt that he couldn???t commit to the intense rigours of club management on a longer-term basis and we respect his honesty.
???In his short time here he was fantastic to work with. He had and will continue to have the utmost respect of everyone here at Sunderland.???